Proof

Proof using the conjugation definition of normality

Proof: This is obvious from the fact that is closed under multiplication.

Proof using the kernel of homomorphism definition of normality

Given: A group .

To prove: There exists a homomorphism for some group such that every element of maps to the identity element.

Proof: Let be the trivial group and be the map sending every element of to the identity element of . Clearly, satisfies the conditions for being a homomorphism: for any , both and equal the identity element of . Moreover, every element of is sent to the identity element of under .

Proof using the cosets definition of normality

Given: A group .

To prove: For every element , .

Proof: Note that

: Clearly, . Also, for any , , so . Thus, .

: Clearly, . Also, for any , , so . Thus, .

Combining the two steps, we obtain that .

Proof using the union of conjugacy classes definition of normality

Given: A group .

To prove: is a union of conjugacy classes in .

Proof: The conjugacy classes form a partition of (arising from the equivalence relation of being conjugate), so is their union.

Proof using the commutator definition of normality

Given: A group .

To prove: For every , .

Proof: This is direct from the fact that is closed under multiplication and inverses, and the commutator is defined in terms of these operations.